如何消除库CVXR中的DCP错误？

Question

我试着用CVXR软件包建立简单的短期销售投资组合模型。下面是众所周知的目标函数(预期收益减去预期收益的sd与已知的teta).但R返回如下：“construct_intermediate_chain中的错误(object，candidate_solvers，gp = gp)：问题不遵循DCP规则。”我试图在下面的代码中修复这个问题，但是没有什么有用的。同样，这种类型的obj函数是凹形的。同时，还可以方便地生成凸函数和求最小值。错误在这种情况下也不会消失。请您解释一下错误在哪里，可能的解决方案是什么。

library(CVXR)
n = 3
w = Variable(n)
c = matrix(1, nrow = 3, ncol = 3)
mu = matrix(1, ncol = 3, nrow = 1)

m1 = rnorm(100)
m2 = rnorm(100)
m3 = rnorm(100)
mu[1, 1] = mean(m1)
mu[1, 2] = mean(m2)
mu[1, 3] = mean(m3)

c[1, 2] = cov(m1, m2)
c[1, 3] = cov(m1, m3)
c[2, 1] = cov(m1, m2)
c[2, 3] = cov(m2, m3)
c[3, 1] = cov(m1, m3)
c[3, 2] = cov(m2, m3)
c[1, 1] = cov(m1, m1)
c[2, 2] = cov(m2, m2)
c[3, 3] = cov(m3, m3)

teta = 0.5
obj1 =  mu %*% w - teta * sqrt(quad_form(w, c))
constr <- list(sum(w) == 1)
pro <- Problem(Maximize((obj)), constr)
sol <- solve(pro)
sol$w

Answer 1

问题是sqrt(quad_form(w, c))：CVXR并不认为它是凸的，尽管它是凸的。您需要计算矩阵c的Cholesky分解，并使用Frobenius范数中的上三角。它相当于sqrt(quad_form(w, c))。

我是说，所以

u = chol(c)

• In

• 计算c的Cholesky分解并得到上三角：norm(u %*% w, "2")

obj1将四边形的平方根替换为norm(u %*% w, "2")

要获得优化后的解决方案，必须使用sol$getValue(w)

您的代码已更新：

> library(CVXR)
> n = 3
> w = Variable(n)
> c = matrix(1, nrow = 3, ncol = 3)
> mu = matrix(1, ncol = 3, nrow = 1)
> 
> m1 = rnorm(100)
> m2 = rnorm(100)
> m3 = rnorm(100)
> mu[1, 1] = mean(m1)
> mu[1, 2] = mean(m2)
> mu[1, 3] = mean(m3)
> 
> c[1, 2] = cov(m1, m2)
> c[1, 3] = cov(m1, m3)
> c[2, 1] = cov(m1, m2)
> c[2, 3] = cov(m2, m3)
> c[3, 1] = cov(m1, m3)
> c[3, 2] = cov(m2, m3)
> c[1, 1] = cov(m1, m1)
> c[2, 2] = cov(m2, m2)
> c[3, 3] = cov(m3, m3)
> 
> teta = 0.5
> u = chol(c)
> obj1 =  mu %*% w - teta * norm(u %*% w, "2")
> constr <- list(sum(w) == 1)
> pro <- Problem(Maximize((obj1)), constr)
> sol <- solve(pro)
> sol$getValue(w)

这就是我查到的：

 > sol$getValue(w)
            [,1]
[1,] -0.07160006
[2,]  0.44897448
[3,]  0.62261508